Variable-Step Multiscale Generalized Gaussian Distribution Generalized Rényi Fuzzy Dispersion Transfer Entropy for Bearing Fault Diagnosis

Dispersion entropy and its transfer-entropy extension are widely used for nonlinear feature extraction in bearing fault diagnosis. Existing improvements address one mechanism at a time, such as fuzzy membership for boundary softening, generalized Gaussian mapping for distribution adaptation, Rényi entropy for probability reweighting, k -step transfer for long-range dependency, or variable-step multiscale coarse-graining for information preservation. No framework combines these mechanisms, and their joint contribution has not been measured. This paper proposes a variable-step multiscale generalized Gaussian distribution generalized Rényi fuzzy dispersion transfer entropy (VSMGGDGRFDTE), in which these modifications act together inside a single dispersion transfer entropy estimator. On three public bearing and motor datasets (CWRU, HUST, and Ottawa) spanning cross-load, cross-speed, and cross-fault-type conditions, and with a support vector machine backbone, VSMGGDGRFDTE attains the highest mean accuracy among the seven entropy methods compared in the progressive-condition experiments on all three datasets: 97.06%, 89.35%, and 93.01%, ahead of the runner-up by 0.80, 3.90, and 2.84 percentage points, respectively, and over 30 shared random partitions every paired comparison remains significant after correction ($p\le 2.0\times {10}^{-5}$). Ablation experiments show that fuzzy membership is the largest single contributor (about 2.2 percentage points of accuracy), followed by variable-step multiscale coarse-graining. At $N=2048$, a single call costs about 18 times the runtime of dispersion transfer entropy, which is acceptable in offline settings where accuracy and consistency matter more than latency.